Asymptotic theory for estimating drift parameters in the fractional Vasicek model

Published in Econometric Theory, Volume 35, Issue 1, February 2019 , pp. 198-231. https://doi.org/10.1017/S0266466618000051</p

Asymptotic Theory for Estimating the Persistent Parameter in the Fractional Vasicek Model

Published in Econometric Theory, Volume 35, Issue 1, February 2019 , pp. 198-231. https://doi.org/10.1017/S0266466618000051</p

Parameter estimation for stochastic differential equations driven by fractional Brownian motion

This dissertation systematically considers the inference problem for stochastic differential equations (SDE) driven by fractional Brownian motion. For the volatility parameter and Hurst parameter, the estimators are constructed using iterated power variations. To prove the strong consistency and the central limit thoerems of the estimators, the asymptotics of the power variatons are studied, which include the strong consistency, central limit theorem, and the convergence rate for the iterated power variations of the Skorohod integrals with respect to fractional Brownian motion. The iterated logarithm law of the power variations of fractional Brownian motion is proved. The joint convergence along different subsequence of power variations of Skorohod integrals is also studied in order to derive the central limit theorem for the estimators. Another important topic considered in this dissertation is the estimation of drift parameters of the SDEs. A least squares estimator (LSE) is proposed and the strong consistency is proved for the fractional Ornstein-Uhlenbeck process that is the solution to the linear SDE. The fourth moment theorem is applied to obtain the central limit theorems. Then the LSE is considered for the drift parameter of the multi-dimensional nonlinear SDE. While proving the strong consistency of LSE, the regularity structure of the SDE’s solution is explored and the maximal inequality for the Skorohod integrals is derived. The main tools used in this research are Malliavin calculus and some Gaussian analysis elements

Statistical inference in high-dimensional matrix models

Matrix models are ubiquitous in modern statistics. For instance, they are used in finance to assess interdependence of assets, in genomics to impute missing data and in movie recommender systems to model the relationship between users and movie ratings. Typically such models are either high-dimensional, meaning that the number of parameters may exceed the number of data points by many orders of magnitudes, or nonparametric in the sense that the quantity of interest is an infinite dimensional operator. This leads to new algorithms and also to new theoretical phenomena that may occur when estimating a parameter of interest or functionals of it or when constructing confidence sets. In this thesis, we will exemplarily consider three such matrix models and develop statistical theory for them: Matrix completion, Principal Component Analysis (PCA) with Gaussian data and transition operators of Markov chains. \\ \\ We start with matrix completion and investigate the existence of adaptive confidence sets in the 'Bernoulli' and 'trace-regression' models. In the 'Bernoulli' model we show that adaptive confidence sets do not exist when the variance of the errors is unknown, whereas we give an explicit construction in the ’trace-regression’ model. Finally, in the known variance case, we show that adaptive confidence sets do also exist in the 'Bernoulli' model based on a testing argument. \\ \\ Next, we consider PCA in a Gaussian observation model with complexity measured by the effective rank, the reciprocal of the percentage of variance explained by the first principal component. We investigate estimation of linear functionals of eigenvectors and prove Berry-Essen type bounds. Due to the high-dimensionality of the problem we discover a new phenomenon: The plug-in estimator based on the sample eigenvector can have non-negligible bias and hence may be not $\sqrt{n}$-consistent anymore. We show how to de-bias this estimator, achieving $\sqrt{n}$-convergence rates, and prove exact matching minimax lower bounds. \\ \\ Finally, we consider nonparametric estimation of the transition operator of a Markov chain and its transition density. We assume that the singular values of the transition operator decay exponentially. For example, this assumption is fulfilled by discrete, low frequency observations of periodised, reversible stochastic differential equations. Using penalization techniques from low rank matrix estimation we develop a new algorithm and show improved convergence rates.Financial support of ERC grant UQMSI/647812 and EPSRC grant EP/L016516/

Change-point Problem and Regression: An Annotated Bibliography

The problems of identifying changes at unknown times and of estimating the location of changes in stochastic processes are referred to as the change-point problem or, in the Eastern literature, as disorder . The change-point problem, first introduced in the quality control context, has since developed into a fundamental problem in the areas of statistical control theory, stationarity of a stochastic process, estimation of the current position of a time series, testing and estimation of change in the patterns of a regression model, and most recently in the comparison and matching of DNA sequences in microarray data analysis. Numerous methodological approaches have been implemented in examining change-point models. Maximum-likelihood estimation, Bayesian estimation, isotonic regression, piecewise regression, quasi-likelihood and non-parametric regression are among the methods which have been applied to resolving challenges in change-point problems. Grid-searching approaches have also been used to examine the change-point problem. Statistical analysis of change-point problems depends on the method of data collection. If the data collection is ongoing until some random time, then the appropriate statistical procedure is called sequential. If, however, a large finite set of data is collected with the purpose of determining if at least one change-point occurred, then this may be referred to as non-sequential. Not surprisingly, both the former and the latter have a rich literature with much of the earlier work focusing on sequential methods inspired by applications in quality control for industrial processes. In the regression literature, the change-point model is also referred to as two- or multiple-phase regression, switching regression, segmented regression, two-stage least squares (Shaban, 1980), or broken-line regression. The area of the change-point problem has been the subject of intensive research in the past half-century. The subject has evolved considerably and found applications in many different areas. It seems rather impossible to summarize all of the research carried out over the past 50 years on the change-point problem. We have therefore confined ourselves to those articles on change-point problems which pertain to regression. The important branch of sequential procedures in change-point problems has been left out entirely. We refer the readers to the seminal review papers by Lai (1995, 2001). The so called structural change models, which occupy a considerable portion of the research in the area of change-point, particularly among econometricians, have not been fully considered. We refer the reader to Perron (2005) for an updated review in this area. Articles on change-point in time series are considered only if the methodologies presented in the paper pertain to regression analysis

Statistical Consequences of Fat Tails: Real World Preasymptotics, Epistemology, and Applications

The monograph investigates the misapplication of conventional statistical techniques to fat tailed distributions and looks for remedies, when possible. Switching from thin tailed to fat tailed distributions requires more than "changing the color of the dress". Traditional asymptotics deal mainly with either n=1 or $n=\infty$, and the real world is in between, under of the "laws of the medium numbers" --which vary widely across specific distributions. Both the law of large numbers and the generalized central limit mechanisms operate in highly idiosyncratic ways outside the standard Gaussian or Levy-Stable basins of convergence. A few examples: + The sample mean is rarely in line with the population mean, with effect on "naive empiricism", but can be sometimes be estimated via parametric methods. + The "empirical distribution" is rarely empirical. + Parameter uncertainty has compounding effects on statistical metrics. + Dimension reduction (principal components) fails. + Inequality estimators (GINI or quantile contributions) are not additive and produce wrong results. + Many "biases" found in psychology become entirely rational under more sophisticated probability distributions + Most of the failures of financial economics, econometrics, and behavioral economics can be attributed to using the wrong distributions. This book, the first volume of the Technical Incerto, weaves a narrative around published journal articles